How can I assign a key frame to the bevel effect
Yes, you can. You only need to set the key frame from the Bevel field by hovering your mouse over it and hitting I.

You need to use the Bevel modifier and not use ctrl+B in Edit mode. You can find it the right side of your screen, on the Properties panel, if you click on Modifiers (the little wrench icon) the list of available modifiers will appear and you just need to click on Bevel to call the Bevel window, then you hover over the Width field and press I to generate a key frame.

Do that at the beginning of the animation, to set the starting point, then change the value on the Bevel Width field to what you want to have at the end of the animation, make sure you are at the end frame as well and hit I again while your mouse is hovering over the Bevel field.